Optimal. Leaf size=139 \[ -\frac{224967}{65219 \sqrt{1-2 x}}+\frac{33115}{1694 \sqrt{1-2 x} (5 x+3)}-\frac{505}{154 \sqrt{1-2 x} (5 x+3)^2}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{5832}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{153825 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]
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Rubi [A] time = 0.0572228, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {103, 151, 152, 156, 63, 206} \[ -\frac{224967}{65219 \sqrt{1-2 x}}+\frac{33115}{1694 \sqrt{1-2 x} (5 x+3)}-\frac{505}{154 \sqrt{1-2 x} (5 x+3)^2}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{5832}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{153825 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]
Antiderivative was successfully verified.
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Rule 103
Rule 151
Rule 152
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx &=\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1}{7} \int \frac{38-105 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{505}{154 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}-\frac{1}{154} \int \frac{2078-7575 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{505}{154 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{33115}{1694 \sqrt{1-2 x} (3+5 x)}+\frac{\int \frac{36534-298035 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{1694}\\ &=-\frac{224967}{65219 \sqrt{1-2 x}}-\frac{505}{154 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{33115}{1694 \sqrt{1-2 x} (3+5 x)}-\frac{\int \frac{-2756361+\frac{3374505 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{65219}\\ &=-\frac{224967}{65219 \sqrt{1-2 x}}-\frac{505}{154 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{33115}{1694 \sqrt{1-2 x} (3+5 x)}-\frac{8748}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{769125 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{2662}\\ &=-\frac{224967}{65219 \sqrt{1-2 x}}-\frac{505}{154 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{33115}{1694 \sqrt{1-2 x} (3+5 x)}+\frac{8748}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{769125 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2662}\\ &=-\frac{224967}{65219 \sqrt{1-2 x}}-\frac{505}{154 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{33115}{1694 \sqrt{1-2 x} (3+5 x)}+\frac{5832}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{153825 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331}\\ \end{align*}
Mathematica [C] time = 0.0401247, size = 78, normalized size = 0.56 \[ \frac{-15524784 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+15074850 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+\frac{77 \left (496725 x^2+612520 x+188306\right )}{(3 x+2) (5 x+3)^2}}{130438 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 91, normalized size = 0.7 \begin{align*} -{\frac{54}{49}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}+{\frac{5832\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{32}{65219}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{31250}{1331\, \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{5}{2} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1353}{250}\sqrt{1-2\,x}} \right ) }-{\frac{153825\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.79637, size = 185, normalized size = 1.33 \begin{align*} \frac{153825}{29282} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2916}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{16872525 \,{\left (2 \, x - 1\right )}^{3} + 75360510 \,{\left (2 \, x - 1\right )}^{2} + 168127762 \, x - 84090985}{65219 \,{\left (75 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 505 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 1133 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 847 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64142, size = 513, normalized size = 3.69 \begin{align*} \frac{52761975 \, \sqrt{11} \sqrt{5}{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 85386312 \, \sqrt{7} \sqrt{3}{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (33745050 \, x^{3} + 24742935 \, x^{2} - 8019782 \, x - 6400750\right )} \sqrt{-2 \, x + 1}}{10043726 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.78419, size = 182, normalized size = 1.31 \begin{align*} \frac{153825}{29282} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2916}{343} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2 \,{\left (215526 \, x - 107875\right )}}{65219 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} - \frac{125 \,{\left (625 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1353 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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